The North Shore is definitely worth a visit and should be one of the items on your list if you rent a car! There is great food in the town of Haleiwa including the famous (and worthwhile) Matsumoto Shave Ice and Giovanni's Shrimp truck. There are also great, well-known snorkling spots up here though we can't vouch for any personally. We've only driven part of the Kamehameha Highway and it is beautiful with hiking spots and beaches along the way. We recommend stopping at the Leonard's Malasadamobile Wagon either to or from the North Shore - you'll get the same famous malasada's as the better known location, but with no line! (Cory & Jaclyn both prefer the original flavor with no filling.)
Whether you can swim or not, we recommend taking surf lessons! We've gone to Surfer Girl Academy twice now and have fun every time. Jaclyn likes them because they're women-owned and super nice. They make you feel really comfortable on the board, help you paddle (which is a lot harder than you expect!!), and you'll get lots of opportunity to catch a wave. They have numerous people around watching so if you're water-shy you can feel safe. No matter where you take lessons though, you'll have fun and we highly recommend giving it a try!
You can, of course, find a place to get SCUBA certified but if you just want to give it a go one time consider booking through Try Scuba Diving. They take you to an enclosed lagoon area and instructors give a lesson staying with you the entire time helping ensure you're safe and comfortable. Jaclyn was VERY nervous the first time and afterward decided she wants to officially get certified. It was an amazing experience and there's nothing quite like it!
There is so much gorgeous and amazing scenery in Hawai'i. Most of our hiking has been on other islands, but in Oahu we've done the Wiliwilinui Ridge Trail and it was absolutely spectacular. It's everything we like in a hike - challenging but not impossible, good length (not too short but not overly long ...though it may feel that way after every summit when you ask "is this the top yet?!"), and a stunning view from the summit. You can take an Uber here (or go on a day when you rented a car) but make sure the Uber actually takes you to the start of the trailhead. They need to pass through the gate into the gated community and continue upward. You'll have the benefit of seeing some of the most coveted real estate on the island (apparently owned by famous people). Bring A LOT of water and wear hiking shoes! I'd recommend bringing a few musubi or other snacks to enjoy when you get to the summit - you'll want to spend some time up there enjoying the 360 view.
We recommend doing a Luau as it's a fun, quintessentially Hawaiian experience. We've done some research to support local institutions and indigenous populations and ended up going to Experience Nutridge. It's a multi-hour experience complete with the history of the estate/Hawaii, dances, fire dancing, gorgeous views, and a delicious dinner experience with show. If you decide to enjoy a different luau, please avoid supporting the Polynesian Cultural Center.
The Ala Moana Center is a MASSIVE outdoor shopping mall with every brand you can imagine. Hawaii has lower taxes than many of the mainland states, so if you have any larger purchases in mind it may be a good idea to spend time here and do some shopping! You can definitely enjoy a couple hours at this open air mall even if it's just window shopping. There are also multiple food courts - one of them with Japanese fruit sandwiches at Mana Sandwiches that are delicious and worth a try if you've never had a fruit sando before.
The Bishop Museum is a must-visit for those interested in learning about the history and environment of Hawai'i. Jaclyn really enjoyed this (but is a museum-enjoyer) and thinks it's an important place for visitors to see. Depending on how much you want to explore, you can spend hours here. It's also near Helena's Hawaiian Food, so you can enjoy Hawaii's famous Kalua Pig, poi, and coconut jelly afterward. Depending on your transportation method you'll also be near enough to make a stop at Jaclyn's favorite place in Oahu - Nisshodo Candy Store (a mochi shop where we bought literally one of every flavor last time).
We highly recommend visiting other islands if you're doing an extended trip! We've been to Maui and the Big Island (the Island of Hawai'i) so can give recommendations individually if you decide to visit. There are affordable flights to and from the islands daily (we even did a day trip to Maui last time!). There is SO MUCH to do on the other islands including Mauna Kea, hiking to gorgeous waterfalls, snorkeling, some of the best food on the islands, literal live lava flows, mountain biking down a volcano, and touring Kona coffee plantations.